calculus of variations geodesic problems

Calculus of variations looks at changes in an entire function, while ordinary calculus only looks at changes in the values of a single function. While calculus of variations is fundamentally based on the same mathematical tools as ordinary calculus (derivatives and integrals), they still differ from one another.Ĭalculus of variations differs from ordinary calculus by considering the optimization of objects called functionals as opposed to ordinary functions. In particular, optimization of a quantity called the action – more on this later!īelow, you’ll find more examples of where calculus of variations is used for. Optimization problems are hugely important in physics as well.Īctually, it turns out that solving for the trajectory of an object under a force is exactly an optimization problem. The main reason why calculus of variations is so important comes from the first thing mentioned above – solving optimization problems.Īn optimization problem is simply a problem in which we want to maximize or minimize a particular quantity or mathematical object (possibly under some constraints as well).Īn example would be to find the biggest margin of profit for a company, given a particular fixed budget – this is an optimization problem that could use the tools of variational calculus. In physics, calculus of variations is especially important for Lagrangian mechanics. This is because it can be used to solve complicated optimization problems, which are important in all of modern science. This will also work as motivation for you to hopefully want to learn more about it.Ĭalculus of variations is an important topic in both physics and mathematical analysis. In this article, you’ll learn both why it’s useful as well as what it’s actually about.įirst, we’ll go over why calculus of variations is important. We’ll also discuss some reasons why calculus of variations is important as well as how it’s applied in physics, geometry and other mathematical contexts.Īs it turns out, calculus of variations is one of the most important areas of math used in physics, yet not many people know much about it. In this article, we’ll look at exactly what these functionals are, what it means to optimize one and how calculus of variations is different from ordinary calculus you may already be familiar with. Calculus of variations can be used, for example, to find the shortest path on a surface or in physics, to describe the motion of a relativistic particle under gravity. However, there is another area of mathematics called calculus of variations that is particularly important for physics.Ĭalculus of variations is the area of mathematics concerned with optimizing mathematical objects called functionals. Most people have heard of basic calculus and know about its applications to, for example, geometry or physics.

Calculus of variations geodesic problems